"He forgot there was an important meeting today."
I'm fairly certain "他今天忘了有重要的會議。" (with the date preceding the verb) should also be accepted.
This sentence seems like it is acceptable in terms of phrasing. However, it is close to "Today, he forgot there was an important meeting". What you are doing with 今天 is that the time of the "meeting" is not indicated, so that the meeting can take place in any of the past time, say yesterday or last week. Instead, you are making "he" does the job "today", forgetting.
Do be careful of how you combine characters together since in practice, Chinese natives are often prosy and descriptive with sentences.